Take a number, such as 25. Now subtract the odds in sequence:
25 - 1 = 24
24 - 3 = 21
21 - 5 = 16
16 - 7 = 9
9 - 9 = 0 [Stop]
It took five subtractions to reach zero, thus the square root of 25 is 5.
Check....does this process work with 16? 9? 225? etc.
Task #1: Show visually why this process works.
Task #2:Can you adjust the process for numbers that are not perfect squares? For example, can you use it (in modified form) to find the square root of 21?
Hint: Draw a five by five square of dots to represent the number 25. Then...
Solution Commentary: Look up what a gnomen is....and how it relates to a carpenter's square.
In the 5 x 5 square of dots, remove 1 dot (i.e. upper left hand corner dot).
Now, remove 3 dots in the shape of a carpenter's square or gnomen...continue with 5 dots, etc.
Do you think this process implies that every square is the sum of consecutive odds? Can you prove it?